Pierre de Fermat

Pierre de Fermat (Beaumont-de-Lomagne, France, August 17, 1601 - Castres, France, January 12, 1665) was a French jurist and mathematician named by the historian of Scottish mathematician, Eric Temple Bell, nicknamed "Prince of Hobbyists". According to Ian Stewart in his book From Here to Infinity, Criticism, 2005, p. 39 . and Singh (2007, p. 57) the nickname was given by Bell himself.

Fermat was, along with René Descartes and Johannes Kepler, one of the leading mathematicians of the first half of the 17th century.

Joseph-Louis Lagrange clearly stated that he considered Fermat to be the inventor of calculus. Fermat co-founded probability theory with Blaise Pascal and independently of Descartes, discovered the fundamental principle of analytic geometry. However, he is best known for his contributions to number theory, especially for what is known as Fermat's Last Theorem, which concerned mathematicians for approximately 350 years, until it was proved in 1995 by Andrew Wiles with the help of Richard Taylor on the basis of the Shimura–Taniyama theorem.

Biography

Fermat was born in the first decade of the 17th century century in Beaumont-de-Lomagne, France; the late-15th century century mansion where Fermat was born is now a museum. He was originally from Gascogne, where his father, Dominique Fermat (a wealthy leather merchant) served three one-year terms as one of Beaumont-de-Lomagne's four consuls. His mother was named Claire de Long.Pierre had one brother and two sisters and almost certainly grew up in his hometown. There is little evidence about his school education, but he probably attended the College of Navarre in Paris in Montauban.

Bust in the Room Henir-Martin at the Capitol of Toulouse

He attended the University of Orleans from 1623 and received a bachelor's degree in civil law in 1626, before going to Bordeaux, where he began his first serious mathematical investigations, and in 1629 gave a copy of his review of the work of Apollonius De Locis Planis to one of the local mathematicians. There is evidence that in Bordeaux he was in contact with Jean de Beaugrand, and during this time he produced an important work on the extrema of a function, which he delivered to Étienne d'Espagnet, who clearly shared mathematical interests with Fermat.. There he was greatly influenced by the work of François Viète.

In 1630, he bought the office of an alderman in the Parliament of Toulouse, one of the high courts of the Judiciary in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life. Thus, Fermat was entitled to change his name from Pierre Fermat to Pierre de Fermat . A fluent speaker in six languages (French, Latin, Occitan, Classical Greek, Italian, and Spanish), Fermat was praised for his verses written in various languages and his advice was frequently sought regarding the revision of Greek texts.

He communicated most of his work in letters to friends, often with little or no proof of his theorems. In some of these letters to his friends, he explored many of the fundamental ideas of calculus before Newton or Leibniz. Fermat was a skilled lawyer who made mathematics more of a hobby than a profession. However, he made important contributions to analytic geometry, probability, number theory, and calculus. Secrecy was common in European mathematical circles at the time. This naturally led to disputes about the priority of some discoveries with his contemporaries, such as Descartes and Wallis.

Anders Dahl writes that "Fermat's mathematics was based on classical Greek treatises combined with the methods of François Viète".

Pierre de Fermat died on January 12, 1665 in Castres, in the present-day department of Tarn.

Mathematical Works

Fermat's spiral

Also known as a parabolic spiral, it is a curve that responds to the following equation in polar coordinates:

r=± ± θ θ 1/2{displaystyle r=pm theta ^{1/2},}

It is a particular case of the Archimedean spiral.

Friendly numbers

Two friendly numbers are two natural numbers a and b such that a is the sum of the divisors proper of b, and b is the sum of proper divisors of a. (The unit is considered a proper divisor, but the number itself is not.)

In 1636, Fermat discovered that 17,296 and 18,416 were a pair of friendly numbers, as well as rediscovering a general formula for calculating them, known to Tabit ibn Qurra, around the year 850.

Prime numbers

A Fermat number is a natural number of the form:

Fn=22n+1{displaystyle F_{n}=2^{2^{n}}+1}

where n is natural.

Pierre de Fermat conjectured that all natural numbers of this form with n natural were prime numbers, but Leonhard Euler proved that this was not the case in 1732. Indeed, by taking n=5 yields a composite number:

F5=225+1=232+1=4294967297=641⋅ ⋅ 6700417{displaystyle F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641cdot 6700417;}

Theorem about the sum of two squares

The Sum of Two Squares Theorem states that every prime number p, such that p-1 is divisible by 4, can be written as a sum of two squares. The 2 is also included, since 1²+1²=2. Fermat announced his theorem in a letter to Marin Mersenne dated December 25, 1640, which is why it is also known as Fermat's Christmas Theorem

Fermat's Little Theorem

Fermat's little theorem, referring to the divisibility of numbers, states that if a number is raised a to the p- and a is subtracted from the result, what remains is divisible by p, where p is a prime number with a and p are coprime. His main interest is in its application to the problem of primality and in cryptography.

Fermat's Principle

Fermat's Last Theorem

Pierre de Fermat used to write the solutions to problems in the margin of books. One of the notes he wrote in his copy of the Greek text of Diophantus of Alexandria's Arithmetica (edited by Claude Gaspard Bachet de Méziriac in 1621) reads as follows:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratosquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginalis exiguitas non caperet.

It is impossible to find a way to convert a cube into the sum of two cubes, a fourth power in the sum of two fourth powers, or in general any higher power than the square, in the sum of two powers of the same class. I've discovered for the fact an excellent demonstration. But this margin is too small for (the demonstration) to fit into it.

Pierre de Fermat

This statement, later known as Fermat's Last Theorem, became one of the most important theorems in mathematics. It is not known if Fermat actually found the proof, since he left no trace of it so that other mathematicians could verify it. This mathematical problem kept mathematicians on edge for more than three centuries (in frustration, Euler is said to have even asked a friend to search Fermat's house for the proof), until in 1995 Andrew Wiles helped by Richard Lawrence Taylor was able to prove the theorem. Wiles used for this mathematical tools that emerged long after Fermat's death, so Fermat must have found the solution in another way, if he did. In any case, he was right.

Way of work

A scholarly man steeped in classical Greco-Roman culture, he was encyclopedic due to the breadth of his baggage. He made notes in the margins of the books he read, with observations and sketches of proofs. He was not a professional mathematician nor did he write books. It was of his interest the human knowledge of his time. He sends letters about his findings or concerns, he had Father Mersenne as a mentor and disseminator, and, instead of formalizing his discoveries or inventions, he possibly dedicated himself to speculating and gave flight to his overflowing imagination; he threw challenges with problems whose solutions he possessed. He argued with Descartes about the case of La Dioptrique his work. Faced with Descartes' discomfort, Fermat sent a test, making present what mattered most to him was the truth, and that he was not driven by a lust for fame or envy.

Acknowledgments

• Fermat is one of the few mathematicians honored as an eponymous of an asteroid, which carries the nominal specification of (12007) Fermat. It has also been given the name of Fermat to a lunar crater of 39 km in diameter.
• The oldest and most prestigious school in Toulouse is called Pierre de Fermat and it offers engineering and trade classes. It is located among the top ten in France for preparatory classes.
• The French sculptor Théophile Barrau made a marble statue called Hommage à Pierre Fermat as a tribute to Fermat, currently at the Capitol of Toulouse.